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Full text of "Analytical Mechanics"

ANGULAR IMPULSE AND ANGULAR MOMENTUM    279
9.   A hoop barely completes rotations about a horizontal axis which passes through its rim and is perpendicular to its plane.    Determine the reaction of the axis at the lowest and the highest positions.
10.   A uniform rod which rotates about a horizontal axis through one end has four times as much kinetic energy as it has potential energy at the instant it passes the highest point.   Find the reaction of the axis when the rod is
(a)  at the highest position;
(b)  horizontal;
(c)  at the lowest position.
215. Impulsive Reaction of an Axis. Center of Percussion. If a rigid body which is free to rotate about a fixed axis is so struck that no impulse is imparted to the axis during the blow, any point of the line of action of the blow is called a center of percussion for that axis. It is evident that if the axis be removed and the blow applied at a center of percussion which corresponds to the removed axis, the body will rotate as if the axis were not removed. The axis about which a free rigid body rotates when it is given a blow is called the axis of spontaneous rotation.
Suppose the rigid body of Fig. 127 to be free to rotate about an axis through 0 perpendicular to the plane of the figure. For the sake of simplicity suppose the blow to be applied in such a direction that it tends to produce rotation only about the given axis. Let L denote the linear impulse of the blow and L'
the impulse given to the body by the reaction of the axis of rotation. Then by the conservation of linear momentum the linear momentum of the body must be equal to the impulse given to it -by the blow and by the reaction of the axis. Therefore
(1)
FIG. 127.